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SUMMARY:Izabella Stuhl (Penn State University)
DTSTART:20200518T120000Z
DTEND:20200518T130000Z
DTSTAMP:20260423T024025Z
UID:HSPETDS/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/HSPETDS/6/">
 The hard-core model in discrete 2D</a>\nby Izabella Stuhl (Penn State Univ
 ersity) as part of Horowitz seminar on probability\, ergodic theory and dy
 namical systems\n\n\nAbstract\nThe hard-core model describes a system of n
 on-overlapping identical hard spheres in a space or on a lattice (more gen
 erally\, on a graph). An interesting open problem is: do hard disks in a p
 lane admit a unique Gibbs measure at high density? It seems natural to app
 roach this question by possible discrete approximations where disks must h
 ave the centers at sites of a lattice or vertices of a graph.\n\nIn this t
 alk\, I will report on progress achieved for the models on a unit triangul
 ar lattice $\\mathbb{A}_2$\, square lattice $\\mathbb{Z}^2$ and a honeycom
 b graph $\\mathbb{H}_2$ for a general value of disk diameter $D$ (in the E
 uclidean metric). We analyze the structure of Gibbs measures for large fug
 acities (i.e.\, high densities) by means of the Pirogov-Sinai theory and i
 ts modifications. It connects extreme Gibbs measures with dominant ground 
 states.\n\nOn $\\mathbb{A}_2$ we give a complete description of the set of
  extreme Gibbs measures\; the answer is provided in terms of the prime dec
 omposition of the Löschian number $D^2$ in the Eisenstein integer ring. O
 n $\\mathbb{Z}^2$\, we work with Gaussian numbers. Here we have to exclude
  a finite collection of values of $D$ with sliding\; for the remaining exc
 lusion distances the answer is given in terms of solutions to a discrete m
 inimization problem. The latter is connected to norm equations in the cycl
 otomic integer ring $\\mathbb{Z}[\\zeta]$\, where $\\zeta$ is a primitive 
 12th root of unity. On $\\mathbb{H}_2$\, we employ connections with the mo
 del on $\\mathbb{A}_2$\, although there are some exceptional values requir
 ing a special approach.\n\nParts of our argument contain computer-assisted
  proofs: identification of instances of sliding\, resolution of dominance 
 issues. This is a joint work with A. Mazel and Y. Suhov.\n
LOCATION:https://researchseminars.org/talk/HSPETDS/6/
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